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Intro Lin Alg Elimination problem

  1. Jun 11, 2010 #1
    The problem statement, all variables and given/known data
    Look for a matrix that has row sums 4 and 8, and column sums 2 and x:

    matrix =
    [a, b] -------I don't know how to do latex
    [c, d]

    a + b = 4; a + c = 2;
    c + d = 8; b + d = s;

    The four equations are solvable only if s = ____. Then find two different matrices that have the correct row and column sums. Write down the 4 by 4 system Ax = b with x = (a, b, c, d) and make A triangular by elimination.

    Attempt at a solution:

    I created a matrix from the system of equations given. I started with:

    [1 1 0 0 4]
    [0 0 1 1 8]
    [1 0 1 0 2]
    [0 1 0 1 s]

    -->


    [1 1 0 0 4]
    [0 -1 1 0 -2]
    [0 0 1 1 8]
    [0 0 1 1 (s-2)]

    -->

    [1 1 0 0 4]
    [0 -1 1 0 -2]
    [0 0 1 1 8]
    [0 0 0 0 (s-10)]

    So I got s = 10.
    I can't solve for a, b, c, or d (or can I?)

    What am I not understanding and how do I do this problem? Thanks.

    edit: Thanks LCKurtz, I got it now.
     
    Last edited: Jun 11, 2010
  2. jcsd
  3. Jun 11, 2010 #2

    LCKurtz

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    Homework Helper
    Gold Member

    Sure you can solve for them. Once you assign s =10, the last equation becomes dependent on the others. You have more variables than equations so you would expect infinitely many solutions. Let d = d (anything) and solve for the others in terms of d by working back up the system. For example your third equation says c + d = 8 so c = 8 - d. So b = ... etc. Make sure you don't have an arithmetic mistake in your reduction (I think you do.) Once you have a, b, and c in terms of d, check that they work in your system and you can continue with the problem.
     
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